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x*(sin(x)^x)
Derivada de la función/ sin(x)/ (x)^x/ x*(sin(x)^x)

Derivada de x*(sin(x)^x)

Función f() - derivada -er orden en el punto
v

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
     x   
x*sin (x)
xsin⁡x(x)x \sin^{x}{\left(x \right)} 
x*sin(x)^x
Solución detallada

  1. Se aplica la regla de la derivada de una multiplicación:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Según el principio, aplicamos: xx tenemos 11

    g(x)=sin⁡x(x)g{\left(x \right)} = \sin^{x}{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. No logro encontrar los pasos en la búsqueda de esta derivada.

      Perola derivada

      xx(log⁡(x)+1)x^{x} \left(\log{\left(x \right)} + 1\right)

    Como resultado de: xxx(log⁡(x)+1)+sin⁡x(x)x x^{x} \left(\log{\left(x \right)} + 1\right) + \sin^{x}{\left(x \right)}

  2. Simplificamos:

    xx+1(log⁡(x)+1)+sin⁡x(x)x^{x + 1} \left(\log{\left(x \right)} + 1\right) + \sin^{x}{\left(x \right)}

Respuesta:

xx+1(log⁡(x)+1)+sin⁡x(x)x^{x + 1} \left(\log{\left(x \right)} + 1\right) + \sin^{x}{\left(x \right)}

Gráfica
02468-8-6-4-2-1010-2000000000010000000000
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Trazar el gráfico f'(x)
Primera derivada [src] 
   x           x    /x*cos(x)              \
sin (x) + x*sin (x)*|-------- + log(sin(x))|
                    \ sin(x)               /
x(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))sin⁡x(x)+sin⁡x(x)x \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)} + \sin^{x}{\left(x \right)}
Simplificar
Segunda derivada [src] 
        /                  /                            2                   2   \             \
   x    |                  |    /x*cos(x)              \    2*cos(x)   x*cos (x)|   2*x*cos(x)|
sin (x)*|2*log(sin(x)) - x*|x - |-------- + log(sin(x))|  - -------- + ---------| + ----------|
        |                  |    \ sin(x)               /     sin(x)        2    |     sin(x)  |
        \                  \                                            sin (x) /             /
(−x(x+xcos⁡2(x)sin⁡2(x)−(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))2−2cos⁡(x)sin⁡(x))+2xcos⁡(x)sin⁡(x)+2log⁡(sin⁡(x)))sin⁡x(x)\left(- x \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) + \frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)}
Simplificar
Tercera derivada [src] 
        /                                 2     /                             3        2                                 /                    2   \          3                \                     2   \
   x    |         /x*cos(x)              \      |     /x*cos(x)              \    3*cos (x)     /x*cos(x)              \ |    2*cos(x)   x*cos (x)|   2*x*cos (x)   2*x*cos(x)|   6*cos(x)   3*x*cos (x)|
sin (x)*|-3*x + 3*|-------- + log(sin(x))|  + x*|-3 + |-------- + log(sin(x))|  - --------- - 3*|-------- + log(sin(x))|*|x - -------- + ---------| + ----------- + ----------| + -------- - -----------|
        |         \ sin(x)               /      |     \ sin(x)               /        2         \ sin(x)               / |     sin(x)        2    |        3          sin(x)  |    sin(x)         2     |
        \                                       \                                  sin (x)                               \                sin (x) /     sin (x)               /                sin (x)  /
(x(2xcos⁡(x)sin⁡(x)+2xcos⁡3(x)sin⁡3(x)+(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))3−3(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))(x+xcos⁡2(x)sin⁡2(x)−2cos⁡(x)sin⁡(x))−3−3cos⁡2(x)sin⁡2(x))−3x−3xcos⁡2(x)sin⁡2(x)+3(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))2+6cos⁡(x)sin⁡(x))sin⁡x(x)\left(x \left(\frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 x \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{3} - 3 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) - 3 - \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - 3 x - \frac{3 x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 3 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} + \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \sin^{x}{\left(x \right)}
Simplificar
Gráfico
Derivada de x*(sin(x)^x)
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